3D Quantum Gravity from Holomorphic Blocks

TL;DR

This paper reformulates 3D quantum gravity using holomorphic blocks derived from spinor-based recoupling symbols, linking quantum geometry to the 2D Ising model and opening new avenues for studying phase transitions and dualities.

Contribution

It introduces a spinor and holomorphic recoupling symbol framework for the Ponzano-Regge model, connecting quantum gravity with 2D conformal theories and Ising model insights.

Findings

Holomorphic recoupling symbols relate to the inverse Ising partition function.

Reformulation enables analysis of phase transitions in 3D quantum gravity.

Establishes a duality framework between 3D quantum gravity and 2D conformal theories.

Abstract

Three-dimensional gravity is a topological field theory, which can be quantized as the Ponzano-Regge state-sum model built from the $\{3nj\}$-symbols of the recoupling of the $\SU(2)$ representations, in which spins are interpreted as quantized edge lengths in Planck units. It describes the flat spacetime as gluing of three-dimensional cells with a fixed boundary metric encoding length scale. In this paper, we revisit the Ponzano-Regge model formulated in terms of spinors and rewrite the quantum geometry of 3D cells with holomorphic recoupling symbols. These symbols, known as Schwinger's generating function for the $\{6j\}$-symbols, are simply the squared inverse of the partition function of the 2D Ising model living on the boundary of the 3D cells. They can furthermore be interpreted, in their critical regime, as scale-invariant basic elements of geometry. We show how to glue them together into a discrete topological quantum field theory. This reformulation of the path integral for 3D quantum gravity, with a rich pole structure of the elementary building blocks, opens a new door toward the study of phase transitions and continuum limits in 3D quantum gravity, and offers a new twist on the construction of a duality between 3D quantum gravity and a 2d conformal theory.